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My question is simply, how many amicable pairs have been found so far using super computers?

I have been trying to find any kind of answer online that is up to date, but after searching in depth I can't find any academic papers (or google results for that matter) that date past 2009.

Have people given up on finding these numbers because they have minimal use in mathematics?

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Other than the search done by Dave Moews in the early 90s I believe that most of the discovered pairs were found on PCs (I know that this is true for most of the pairs less than 10^14). One can easily use the known algorithms to generate a few pairs per PC hour, if that amuses you. I think that there is not much point to finding more general pairs, though looking for a relatively prime pair would be feasible and interesting. –  deinst Feb 14 '12 at 4:51

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There is a list of known amicable pairs, perfect numbers, sociable numbers, etc. here. This is the most up-to-date list of amicable numbers that I know of, although, unfortunately, it has not been updated since September 28, 2007. It lists 11,994,387 amicable pairs. According to the site, the list is exhaustive for $a<10^{14}$ ($a$ being the smaller member of an amicable pair), and there are $39,374$ such pairs.

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Unfortunately that is the same list that I also found. I suppose it is possible that now that so many are found, the only thing left to do with amicable pairs is prove that there exist infinitely many. –  soleimc Feb 14 '12 at 3:21
    
Or else one can look for applications. There is a nice comment by someone long ago who reported having tested, one hopes successfully, the erotic effect of feeding $220$ to his lover while himself eating $284$. –  André Nicolas Feb 14 '12 at 7:10
    
It was al-Majriti, circa $1000$ CE. –  André Nicolas Feb 14 '12 at 7:31

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